Gaphs of Functions with
Maple
Mika Seppälä
Plotting Graphs with Computers (1)
Computers form plots of graphs or curves by computing a number of points
on the graph or curve and then connecting these points to form an
approximation of the graph.
Here is a plot of the curve defined by the equation
 x 2  y 2   3x 2 y  y 3  0.
2
The picture was generated by Maple 8
using its implicitplot command (in the plots
library).
To form the plot, Maple has substituted
numeric values for x in the above
equation and solved all these equations
for y. To form the graph, Maple has then
connected the computed points with
straight lines.
Mika Seppälä: Graphs of Functions
Plotting Graphs with Computers (2)
The plot generated by Maple looks suspicious at the origin. The problem is
that several branches of the graph come to the origin. When Maple has
computed points on the curve then it needs to figure out which points have
to be connected with straight lines to form a correct plot.
A better result can be obtained by using the polar coordinates.
Substituting x  r cos( ), y  r sin( ) to the equation

x2  y 2

2
 3x 2 y  y 3  0
and solving for r leads to the equation
r  sin3    3cos2   sin  
for the curve C.
Mika Seppälä: Graphs of Functions
Plotting Graphs with Computers (4)
Polar coordinate representation gives immediately a parameterization of the
graph. This is much better for plotting, since the parameterization also can
be used to decide which points have to be connected to which points. Here
are the two graphs.
Implicit plot
graph
Mika Seppälä: Graphs of Functions
Graph by a
parameterization
Scaling
When plotting functions it is important to define the viewing rectangle
correctly. Programs have built in methods to choose the viewing rectangle
1
but these methods do not always lead to desired results.
The picture on the right was generated
by Maple. It shows a portion of the
graph of the function sin(50x). Maple
has chosen, by default, to changes
scales. In this picture, x-axis and yaxis use different scales. Hence the
picture is not accurate even though it
may be right for the purposes needed.
0.25
-0.25
-1
f  x  =sin  50x .
Mika Seppälä: Graphs of Functions
Viewing Rectangles – x Axis
Consider again function sin(50x). Here are plots of the graph of this
function for different x-intervals. Some of the plots are clearly incorrect.
Sin(50x), -6x6
Sin(50x), -10x10
Mika Seppälä: Graphs of Functions
Sin(50x), -0.1x0.1
Viewing Rectangles – y Axis
Choosing the height of the viewing rectangle correctly is also important. If
the y-axis range of the viewing rectangle is not specified, then Maple
chooses it so that it can show all of the graph. This is not always a good
choice.
ex+sin(x), -10x10
ex+sin(x), -10x1
Mika Seppälä: Graphs of Functions